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On the Orbital Evolution of a Pair of Giant Planets in Mean Motion Resonance

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On the Orbital Evolution of a Pair of Giant Planets in Mean Motion Resonance MNRAS 461, 4406–4418 (2016) doi:10.1093/mnras/stw1577 Advance Access publication 2016 July 4 On the orbital evolution of a pair of giant planets in mean motion resonance Q. Andre´1,2‹ and J. C. B. Papaloizou1 1DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK 2Departement´ de Physique, ENS Cachan, Universite´ Paris-Saclay, 61 Avenue du President´ Wilson, F-94230 Cachan, France Accepted 2016 June 30. Received 2016 June 30; in original form 2015 July 9 ABSTRACT Pairs of extrasolar giant planets in a mean motion commensurability are common with 2:1 Downloaded from resonance occurring most frequently. Disc–planet interaction provides a mechanism for their origin. However, the time-scale on which this could operate in particular cases is unclear. We perform 2D and 3D numerical simulations of pairs of giant planets in a protoplanetary disc as they form and maintain a mean motion commensurability. We consider systems with http://mnras.oxfordjournals.org/ current parameters similar to those of HD 155358, 24 Sextantis and HD 60532, and disc models of varying mass, decreasing mass corresponding to increasing age. For the lowest mass discs, systems with planets in the Jovian mass range migrate inwards maintaining a 2:1 commensurability. Systems with the inner planet currently at around 1 au from the central star could have originated at a few au and migrated inwards on a time-scale comparable to protoplanetary disc lifetimes. Systems of larger mass planets such as HD 60532 attain 3:1 resonance as observed. For a given mass accretion rate, results are insensitive to the disc model for the range of viscosity prescriptions adopted, there being good agreement between 2D and 3D simulations. However, in a higher mass disc a pair of Jovian mass planets passes through at University of Cambridge on August 10, 2016 2:1 resonance before attaining a temporary phase lasting a few thousand orbits in an unstable 5:3 resonance prior to undergoing a scattering. Thus, finding systems in this commensurability is unlikely. Key words: accretion, accretion discs – hydrodynamics – methods: numerical – planet–disc interactions – protoplanetary discs – planetary systems. figurations in situ is expected to be small (e.g. Beauge, Ferraz-Mello 1 INTRODUCTION & Michtchenko 2012). Pairs of extrasolar giant planets in a mean motion commensurability Disc–planet interaction can produce the required evolution of the are a common occurrence. It has been estimated that sixth of multi- semi-major axes. This may result in convergent migration leading planet systems detected by the radial velocity technique are in or to the formation of a commensurability (see Baruteau et al. 2014 close to a 2:1 commensurability (Wright et al. 2011). Parameters for and references therein). Accordingly, understanding the observed some cases of interest that are considered in this paper are shown configuration of such systems has the potential for either revealing in Table 1 (for additional examples see e.g. Emelyanenko 2012). how disc–planet interactions may have operated or for ruling them In addition, there are two known systems in 3:2 resonance (Correia out. et al. 2009;Reinetal.2012; Robertson et al. 2012b) and one in a Previous numerical studies of commensurabilities forming and 4:3 resonance (Johnson et al. 2011;Reinetal.2012) consistent with evolving as a result of disc–planet interactions have focused on the systems in 2:1 resonance being the most commonly observed systems such as GJ 876, HD 45364 and HD 6805 interacting with commensurability. disc modelled with either constant kinematic viscosity or with the The existence of these resonant systems indicates that dissipative α-viscosity parameter of Shakura & Sunyaev (1973) taken to be con- mechanisms that result in changes to planet semi-major axes that stant (for a review see Baruteau et al. 2014 and references therein). produce related changes to period ratios in planetary systems have In this paper, we extend such studies, considering systems with or- operated. This is because the probability of forming resonant con- bital parameters similar to those of HD 155358 (Robertson et al. 2012a), 24 Sextantis (Johnson et al. 2011), HD 60532 (Laskar & Correia 2009) as well as HD 6805 (Trifonov et al. 2014) each E-mail: [email protected] of which have the inner planet with semi-major axis in the range C 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society A pair of giant planets in MMR 4407 Table 1. Properties of the HD 155358, 24 Sextantis, HD 6805 and HD 2 BASIC EQUATIONS 60532 systems. The first three either are or possibly in 2:1 resonance while θ ϕ the fourth is in a 3:1 resonance. The first column identifies the system, and We adopt a spherical coordinate system (r, , ) with associated the second column gives the mass of the central star in solar masses. The unit vectors (rˆ, θˆ, ϕˆ ) and origin at the centre of mass of the central third and fourth columns give the masses of the planets in Jupiter masses star. and the fifth and sixth columns give their semi-major axes in au. The basic equations governing the disc express the conserva- tion of mass and momentum under the gravitational potential due System M∗ M1 M2 a1 a2 to the central star and any planets and incorporate a kinematic viscosity ν, HD 155358 0.92 0.85 ± 0.05 0.82 ± 0.07 0.64 1.02 24 Sextantis 1.54 1.99 ± 0.4 0.86 ± 0.4 1.33 2.08 ∂ρ =−∇· ρv , HD 6805 1.7 2.5 ± 0.2 3.3 ± 0.2 1.27 1.93 ∂t ( ) (1) HD 60532 1.44 3.15 7.46 0.77 1.58 Dv ρ =−ρ∇ −∇P +∇· T . Dt (2) Here the convective derivative is defined through 0.5–1.4 au. We perform 2D and 3D numerical simulations of pairs D ∂ ≡ + v ·∇, Downloaded from of giant planets interacting with a protoplanetary disc that attain a Dt ∂t (3) mean motion commensurability for up to 2.5 × 104 orbital peri- ods of the inner planet. We investigate whether such systems could ρ is the density, v is the velocity, P is the pressure, T is the viscous have originated at larger radii beyond the ice line and then migrated stress tensor (see e.g. Mihalas & Weibel Mihalas 1984), and is the inwards, the commensurability possibly being formed in the same gravitational potential which has contributions from the central star neighbourhood. and any planets present. Disc self-gravity is neglected. The pressure http://mnras.oxfordjournals.org/ In order to study the role of the nature of the underlying disc is related to the gas density and the isothermal speed of sound cs P = ρc2. model, we consider models with an inner magnetorotational insta- through s bility (MRI) active region producing a significant effective viscosity and an outer inactive region for which a significant effective vis- 3 NUMERICAL SIMULATIONS cosity may occur only in the upper layers of the disc (Gammie 1996) as well as models with a uniform α-viscosity prescription Simulations were performed using the finite volume fluid dynamics throughout. We also consider models with different surface density code PLUTO (Mignone et al. 2007) which has been used success- scaling corresponding to varying the total disc mass or equivalently fully to simulate protoplanetary discs interacting with planets (e.g. the steady-state accretion rate. In this way, the disc–planet interac- Mignone et al. 2012; Uribe, Klahr & Henning 2013). The planet at University of Cambridge on August 10, 2016 tion at different stages of the life of the protoplanetary disc can be positions are advanced using a fourth-order Runge–Kutta method studied with lower mass discs corresponding to later stages (e.g. which however assumes that the forces due to the disc do not change Calvet et al. 2004). as the planet locations are advanced through a time step, making We find that when low-mass disc models are considered, systems the method one of lower order (see e.g. Nelson et al. 2000 for with planets in the Jovian mass range maintain a 2:1 commensu- comparison). For some runs, we employed the FARGO algorithm rability while undergoing inward type II migration. This is found of Masset (2000) as this allows the numerical calculations to run to be at a rate such that formation at a few au from the central significantly faster. We note that for our simulations the options star and migration to their current locations on a time-scale com- were chosen such that algorithm was applied using the residual parable to the expected protoplanetary disc lifetime is possible in azimuthal velocity with respect to the initial azimuthal velocity, principle. the latter not being updated. A sample of runs were checked care- We find that there is a relative insensitivity of results to the fully to confirm numerical stability and that consistent results were disc model employed and find good agreement between 2D and 3D obtained (see also Mignone et al. 2012 for a comparison of this simulations. Planets containing larger masses such as the HD 60532 type). system which is observed to be in 3:1 resonance (Laskar & Correia For the calculations reported here, we adopt a locally isother- −1/2 2009) are found to attain this resonance in low-mass low-viscosity mal equation of state for which cs ∝ r . The constant of pro- discs. portionality is chosen so as to give a constant aspect ratio h ≡ We find that for systems with planets in the Jovian mass range, H/r = cs/(r K) = 0.05. Here H is the disc semi-thickness and K increasing the disc mass results in the formation of an unstable is the local Keplerian circular orbit angular velocity.
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